Automotive Cornering & Tire Traction Limits

In automotive and autonomous vehicle engineering, electronic stability control (ESC) systems continuously monitor vehicle kinematics to prevent skidding. By decomposing the total acceleration vector into its tangential component $a_T$ (longitudinal drive torque/braking) and normal component $a_N$ (lateral cornering grip), vehicle computers compare lateral acceleration against the friction limit $a_{\text{grip}} = \mu_s g$ of the tire contact patch.

An autonomous electric vehicle prototype is undergoing track testing at the Las Vegas Motor Speedway. Its 2D trajectory across the asphalt is tracked in meters as a function of time $t$ (in seconds) by high-precision RTK-GPS:

\vec{r}(t) = \langle t^3 - 3t, 2t^2, 0 \rangle

The tires provide a maximum lateral (normal) acceleration threshold of

a_{N,\max} = 12\text{ m/s}^2

before static friction breaks and the vehicle enters an uncontrolled lateral slide.

State the formulas for the tangential component $a_T$ and normal component $a_N$ of acceleration.
Compute the exact velocity $\vec{v}(2)$ , acceleration $\vec{a}(2)$ , and speed $\|\vec{v}(2)\|$ at time $t = 2\text{ seconds}$ .
Calculate the tangential acceleration $a_T$ and normal acceleration $a_N$ at $t = 2\text{ seconds}$ .
Determine if the vehicle maintains tire traction or if it exceeds the critical skid limit at $t = 2\text{ seconds}$ .

Tony's Intertwingled Memex • Reference Manual

Automotive Cornering & Tire Traction Limits